Given a fixed p = 2, we prove a simple and effective characterization of all radial multipliers of FL p (R d ), provided that the dimension d is sufficiently large. The method also yields new L q space-time regularity results for solutions of the wave equation in high dimensions.

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We investigate connections between radial Fourier multipliers on R d and certain conical Fourier multipliers on R d+1 . As an application we obtain a new weak type endpoint bound for the Bochner-Riesz multipliers associated with the light cone in R d+1 , where d ≥ 4, and results on characterizations of L p → L p,ν inequalities for convolutions with radial kernels.